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Spherical submanifolds which are of 2-type via the second standard immersion of the sphere

Published online by Cambridge University Press:  22 January 2016

Manuel Barros  and
Bang-Yen Chen
Manuel Barros
Affiliation:
Departamento de Geometria y Topologia University de Granada, 18071—Granada, Spain
Bang-Yen Chen
Affiliation:
Department of Mathematics Michigan State University, East Lansing, Michigan 48824, USA
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Let Sm(r) be an m-sphere of constant sectional curvature 1/r2 and M an n-dimensional compact minimal submanifold of Sm(r). If Sm(r) is imbedded in Em+1 by its first standard imbedding, then, by a well-known result of Takahashi [11], the Euclidean coordinate functions restricted to M are eigenfunctions of Δ on M with the same eigenvalue n/r2. Moreover, the center of mass of M in Em+1 coincides with the center of the hypersphere Sm(r) in Em+1. Thus, M is mass-symmetric in Sm(r) ⊂ Em+\ Consequently, we see that if one wants to study the spectral geometry of a submanifold of Sm(r), it is natural to immerse Sm(r) by its k-th standard immersion, in particular, by its second standard immersion.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 108 , December 1987 , pp. 77 - 91
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1987

References

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